Fostering first-year pre-service teachers’ proof competencies

Abstract

In the study presented, as we report in this paper, we describe our theoretical and practical consideration to engage first-year pre-service teachers in proving activities in the context of a transition-to-proof course. We investigated how students argued to verify a claim of elementary number theory on entering university and compared the results to their performance in the final examination of the course. Subsequently, we elaborate on the following results: On entering university, students do not seem to be capable of using algebraic variables as a heuristic to engage in reasoning. However, after learning about different kinds of proofs and the symbolic language of mathematics, students give evidence of starting to value mathematical language and of enhancing their proof competencies.

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Fig. 1

(Figure similar to Mason and Pimm 1984, p 284)

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Notes

  1. 1.

    In this paper, “reasoning” is used as a superordinate concept that comprises all kinds of possible approaches to verify a claim.

  2. 2.

    The testing of the claim helps the students to understand the claim and to get an idea about its truth [cf. the structure of the proving process given in Boero (1999, p. 2)].

  3. 3.

    Definition 1.1: \(n\in \mathbb{N}\) is divisible by \(a\in \mathbb{N}\), iff there is a number \(q\in \mathbb{N}\) with \(n = a \cdot q\).

  4. 4.

    \({D}_{n}\) is the nth triangular number: \(1+2+3+\dots +n=\frac{n(n+1)}{2}\).

  5. 5.

    In this paper, the word “argumentation” is meant to describe an approach, where a person applies some warrant on data to support a conclusion (in the sense of Toulmin 1958).

  6. 6.

    An explanation for this result may be that we did not use many tasks where students had to make use of two variables to prove a given claim. This problem should be considered in the next implementation of the course.

  7. 7.

    A complete solution would have been for example\(\dots =6n+15=2\cdot \left(3n+7\right)+1.\) Since \(\left(3n+7\right)\in \mathbb{N}\), the result is an odd number (Theorem 2.1).

References

  1. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp 216–235). London: Hodder and Stoughton.

    Google Scholar 

  2. Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23–40.

    Article  Google Scholar 

  3. Biehler, R., & Kempen, L. (2013). Students’ use of variables and examples in their transition from generic proof to formal proof. In B. Ubuz, C. Haser & M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 86–95). Ankara: Middle East Technical University. Retreived from http://cerme8.metu.edu.tr/wgpapers/WG1/WG1_Kempen.pdf. Accessed 12 Sep 2018.

  4. Biehler, R., & Kempen, L. (2016). Didaktisch orientierte Beweiskonzepte—Eine Analyse zur mathematikdidaktischen Ideenentwicklung. Journal für Mathematik-Didaktik, 37(1), 141–179.

    Article  Google Scholar 

  5. Blum, W. (1998). On the role of “Grundvorstellungen” for reality-related proofs: Examples and reflections. Proceedings of CERME 3. Retrieved from http://www.erme.tu-dortmund.de/%7Eerme/CERME3/Groups/TG4/TG4_Blum_cerme3.pdf. Accessed 12 Sep 2018.

  6. Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22, 183–203.

    Article  Google Scholar 

  7. Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof, July/August.

  8. Brunner, E. (2013). Innermathematisches Beweisen und Argumentieren in der Sekundarstufe 1. Mögliche Erklärungen für systematische Bearbeitungsunterschiede und leistungsförderliche Aspekte. Münster: Waxmann.

    Google Scholar 

  9. Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving in the classroom. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI Study (pp. 191–214). Heidelberg: Springer.

    Google Scholar 

  10. Hemmi, K. (2006). Approaching proof in a community of mathematical practice. Stockholm University: Stockholm. Retrieved from http://www.diva-portal.org/smash/get/diva2:189608/FULLTEXT01.pdf. Accessed 18 Sep 2018.

  11. Karunakaran, S., Freeburn, B., Konuk, N., & Arbaugh, F. (2014). Improving preservice secondary mathematics teachers’ capability with generic example proofs. Mathematics Teacher Educator, 2(2), 158–170.

    Article  Google Scholar 

  12. Kempen, L. (2019). Begründen und Beweisen im Übergang von der Schule zur Hochschule. Wiesbaden: Springer Spektrum.

    Google Scholar 

  13. Kempen, L., & Biehler, R. (2014). The quality of argumentations of first-year pre-service teachers. In S. Oesterle, P. Liljedahl, C. Nicol, & D. Allen (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 3, pp. 425–432). Vancouver, Canada: PME.

    Google Scholar 

  14. Kuckartz, U. (2012). Qualitative Inhaltsanalyse. Methoden, Praxis, Computerunterstützung. Weinheim/Basel: Beltz Juventa.

    Google Scholar 

  15. Leikin, R. (2009). Multiple proof tasks: Teacher practice and teacher education. In F. L. Lin, F. J. Hsieh, G. Hanna & M. De Villiers (Eds.), ICMI Study 19: Proof and proving in mathematics education (Vol (2, pp. 31–36). Taipei: The Department of Mathematics, National Taiwan Normal University.

    Google Scholar 

  16. Leron, U., & Zaslavski, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30.

    Google Scholar 

  17. Malle, G. (1993). Didaktische Probleme der elementaren Algebra. Braunschweig/Wiesbaden: Vieweg & Sohn.

    Google Scholar 

  18. Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in Algebra. London: The Open University.

    Google Scholar 

  19. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.

    Article  Google Scholar 

  20. Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematical proof. Educational Studies in Mathematics, 48, 83–99.

    Article  Google Scholar 

  21. Reid, D., & Vallejo Vargas, E. (2018). When is a generic argument a proof? In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective (pp. 239–251). Cham: Springer International Publishing.

    Google Scholar 

  22. Selden, A. (2012). Transitions and proof and proving at tertiary level. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI Study (pp. 391–422). Heidelberg: Springer.

    Google Scholar 

  23. Selden, A., & Selden, J. (2007). Overcoming students’ difficulties in learning to understand and construct proofs. Making the connection: Research and practice in undergraduate mathematics. MAA Notes.

  24. Tall, D. (1979). Cognitive aspects of proof, with special reference to the irrationality of √2. In D. Tall (Ed.), Proceedings of the 3rd International Conference on the Psychology of Mathematics Education (pp. 203–205). Warwick: The Mathematics Education Research Center.

    Google Scholar 

  25. Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  26. Wittmann, E. C. (2009). Operative proof in elementary mathematics. In F.-L. Lin, F.-J. Hsieh, G. Hanna & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education (Vol. 2, pp. 251–256). Taipei: The Department of Mathematics, National Taiwan Normal University.

    Google Scholar 

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Correspondence to Leander Kempen.

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Kempen, L., Biehler, R. Fostering first-year pre-service teachers’ proof competencies. ZDM Mathematics Education 51, 731–746 (2019). https://doi.org/10.1007/s11858-019-01035-x

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Keywords

  • Transition to university
  • Generic proof
  • Variables
  • Proving